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(*         *   The Coq Proof Assistant / The Coq Development Team       *)
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(*                                                                      *)
(* Micromega: A reflexive tactic using the Positivstellensatz           *)
(*                                                                      *)
(*  Frédéric Besson (Irisa/Inria) 2006-2008                             *)
(*                                                                      *)
(************************************************************************)

Require Import List.
Require Setoid.

Set Implicit Arguments.

(* Refl of '->' '/\': basic properties *)

Fixpoint make_impl (A : Type) (eval : A -> Prop) (l : list A) (goal : Prop) {struct l} : Prop :=
  match l with
    | nil => goal
    | cons e l => (eval e) -> (make_impl eval l goal)
  end.

Theorem make_impl_true :
  forall (A : Type) (eval : A -> Prop) (l : list A), make_impl eval l True.
Proof.
intros A eval l; induction l as [| a l IH]; simpl.
- trivial.
- intro; apply IH.
Qed.


Theorem make_impl_map :
  forall (A B: Type) (eval : A -> Prop) (eval' : A*B -> Prop) (l : list (A*B)) r
         (EVAL : forall x, eval' x <-> eval (fst x)),
    make_impl eval' l r <-> make_impl eval (List.map fst l) r.
Proof.
intros A B eval eval' l; induction l as [| a l IH]; simpl.
- tauto.
- intros r EVAL.
  rewrite EVAL.
  rewrite IH.
  + tauto.
  + auto.
Qed.

Fixpoint make_conj (A : Type) (eval : A -> Prop) (l : list A) {struct l} : Prop :=
  match l with
    | nil => True
    | cons e nil => (eval e)
    | cons e l2 => ((eval e) /\ (make_conj eval l2))
  end.

Theorem make_conj_cons : forall (A : Type) (eval : A -> Prop) (a : A) (l : list A),
  make_conj eval (a :: l) <-> eval a /\ make_conj eval l.
Proof.
intros A eval a l; destruct l; simpl; tauto.
Qed.


Lemma make_conj_impl : forall (A : Type) (eval : A -> Prop) (l : list A) (g : Prop),
  (make_conj eval l -> g) <-> make_impl eval l g.
Proof.
  intros A eval l; induction l as [|? l IHl].
  - simpl.
    tauto.
  - simpl.
    intros g.
    destruct l.
    + simpl.
      tauto.
    + generalize (IHl g).
      tauto.
Qed.

Lemma make_conj_in : forall (A : Type) (eval : A -> Prop) (l : list A),
  make_conj eval l -> (forall p, In p l -> eval p).
Proof.
  intros A eval l; induction l as [|? l IHl].
  - simpl.
    tauto.
  - simpl.
    intros H ? H0.
    destruct l.
    + simpl in H0.
      destruct H0.
      * subst; auto.
      * tauto.
    + destruct H.
      destruct H0.
      * subst;auto.
      * apply IHl; auto.
Qed.

Lemma make_conj_app : forall  A eval l1 l2, @make_conj A eval (l1 ++ l2) <-> @make_conj A eval l1 /\ @make_conj A eval l2.
Proof.
  intros A eval l1; induction l1 as [|a l1 IHl1].
  - simpl.
    tauto.
  - intros l2.
    change ((a::l1) ++ l2) with (a :: (l1 ++ l2)).
    rewrite make_conj_cons.
    rewrite IHl1.
    rewrite make_conj_cons.
    tauto.
Qed.

Infix "+++" := rev_append (right associativity, at level 60) : list_scope.

Lemma make_conj_rapp : forall  A eval l1 l2, @make_conj A eval (l1 +++ l2) <-> @make_conj A eval (l1++l2).
Proof.
  intros A eval l1; induction l1 as [|? ? IHl1].
  - simpl. tauto.
  - intros.
    simpl rev_append at 1.
    rewrite IHl1.
    rewrite make_conj_app.
    rewrite make_conj_cons.
    simpl app.
    rewrite make_conj_cons.
    rewrite make_conj_app.
    tauto.
Qed.

Lemma not_make_conj_cons : forall (A:Type) (t:A) a eval  (no_middle_eval : (eval t) \/ ~ (eval  t)),
  ~ make_conj  eval (t ::a) <-> ~  (eval t) \/ (~ make_conj  eval a).
Proof.
  intros.
  rewrite make_conj_cons.
  tauto.
Qed.

Lemma not_make_conj_app : forall (A:Type) (t:list A) a eval
  (no_middle_eval : forall d, eval d \/ ~ eval d) ,
  ~ make_conj  eval (t ++ a) <-> (~ make_conj  eval t) \/ (~ make_conj eval a).
Proof.
  intros A t; induction t as [|a t IHt].
  - simpl.
    tauto.
  - intros a0 **.
    simpl ((a::t)++a0).
    rewrite !not_make_conj_cons by auto.
    rewrite IHt by auto.
    tauto.
Qed.
